Simplify: $\frac{x^3+15x^2+75x+125}{x^2-25}\times (x-5)$

To simplify the given expression, let's write it down first:
$\frac{x^3+15x^2+75x+125}{x^2-25}\times (x-5)$

Let's analyze and factorize the numerator and the denominator of the fraction step-by-step.

Step 1: Factorize the numerator
The numerator is:
$x^3+15x^2+75x+125$
We can recognize this expression as a perfect cube. Recall the algebraic identity for the cube of a binomial:
${(a+b)}^3=a^3+3a^{2}b+3a{b}^2+b^3$
Comparing this with our numerator by letting $a=x$ and $b=5$:
${(x+5)}^3=x^3+3\left(x^2\right)\left(5\right)+3\left(x\right)\left(5^2\right)+5^3$
${(x+5)}^3=x^3+15x^2+75x+125$
Thus, the numerator simplifies to:
${(x+5)}^3$

Step 2: Factorize the denominator
The denominator is:
$x^2-25$
This is a difference of squares, which can be factored using the identity $a^2-b^2=(a-b)(a+b)$:
$x^2-5^2=(x-5)(x+5)$

Step 3: Substitute the factored terms back into the original expression
Substitute the factored forms of the numerator and denominator back into the expression:
$\frac{{(x+5)}^3}{(x-5)(x+5)}\times (x-5)$

Step 4: Cancel out common terms
We can cancel the common factor $(x-5)$ from the numerator and denominator:
$\frac{{(x+5)}^3}{x+5}$
Next, divide ${(x+5)}^3$ by $(x+5)$:
${(x+5)}^2$

Step 5: Expand the simplified expression
Now expand ${(x+5)}^2$ using the identity ${(a+b)}^2=a^2+2ab+b^2$:
$x^2+2\left(x\right)\left(5\right)+5^2$
$x^2+10x+25$

Therefore, the expression simplifies to $x^2+10x+25$.